Curved Surface Area Cylinder Formula

A cylinder is a three-dimensional solid, having a circular base and top that are parallel to each other. The perpendicular distance between the top and the base is known as the height of the cylinder. The surface area of a cylinder is the area surrounded by its surface in a three-dimensional space. Generally, the area of the three-dimensional shapes represents the surface area. The surface area is represented in square units, such as cm², m², and so on. A cylinder can be seen when a set of circular disks are stacked on one another. 

Here, we will discuss the curved surface area of a cylinder and how to find the curved surface area of a cylinder.

Cylinder Area Definition

The area of the cylinder is the total region occupied by a cylinder in the three-dimensional space. The area of the cylinder is obtained by the sum of the area of two circular bases and curved surface area. In the right cylinders, the two circular bases are exactly placed over each other. In the middle of the two circular bases of the cylinder, there is a curved surface, when we open it forms a rectangular shape. The curved surface is also known as the lateral surface. The different parameters that are used to calculate the area of the cylinder include radius, height, axis, base, and side. The radius of the cylinder is the radius of the circular base. The height of the cylinder is the measurement of perpendicular distance between two circular bases.  The line that passes through the centre of the base is called the axis of a cylinder. 

How to Find the Curved Surface Area of a Cylinder?

The curved surface area of a cylinder (also known as CSA) is defined as the area of the curved surface of any given cylinder having base radius ‘r’, and height ‘h’. The curved area is also known as the lateral surface area.

Formula to calculate the curved surface area is given below:

The formula for CSA of cylinder = 2πrh

Derivation of the Curved Surface Area of Cylinder Formula

Here we will derive the formula of the curved surface area of the cylinder. Let’s consider a cylinder of base radius ‘r’ and height ‘h’ units. The curved surface of this cylinder forms a rectangle when we open it along the diameter (d=2r) of the circular base.

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Now, from the above figure, we can observe, the curved surface of the cylinder in two-dimension space forms a rectangle. Hence, the height and circumference of the circular bases are the dimensions of the rectangle that are formed from it. Therefore,

Curved surface area = Height x Circumference

Curved Surface area = h x  πd = h x 2πr (here d = 2r)

Formula for CSA of cylinder = 2πrh 

Hence curved surface area cylinder formula is 2πrh sq. units

Solved Examples:

1. Find the Curved Surface Area of a Cylinder Which Has a Height of 8 cm and a Diameter of the Base is 2 cm.

Ans: Given height (h)= 8 cm and diameter (d)=2 cm

Curved surface area of the cylinder formula is 2πrh

So we have to find the radius value

We know that r = \[\frac{d}{2}\]

So radius (r) = \[\frac{2}{2}\] = 1cm

Put the value of r and h in the formula, we get

Curved surface area = 2π x 1 x 8

                                  = 16 x 3.14

                                   =50.24 cm² 

Therefore, the curved surface area of a cylinder = 50.24 cm² 

2. The Curved Surface Area of a Cylinder is 154 cm² and the Height is Equal to Twice the Radius of the Base. Find the Radius and Height of a Cylinder.

Ans: Given C.S.A. = 154 cm²

h=2r

The curved surface area of a cylinder = 2πrh sq. units

So 2πrh = 154

Now, put the value of h=2r in the above equation

2π x r x 2r = 154

⇒ 2 x \[\frac{22}{7}\] x 2r² = 154

⇒ r² = \[\frac{154 \times 7}{22 \times 4}\] 

⇒ r = \[\sqrt{\frac{49}{4}}\]

⇒ r = \[\frac{7}{2}\] 

⇒ r = 3.5 cm

Height = 2r= 2 x 3.5 = 7cm

Hence height = 7 cm and radius = 3.5 cm